Paper-and-Pencil Skills 'Impede' Math Progress

Breakthrough Made in Superconduction'' declares the headline on the
front page of the morning paper. The story reports the development of a
new ceramic material that can carry electrical current with minute
resistance, at temperatures warm enough to be of practical use. One
possible outcome, the story reports, is that desk-sized supercomputers
could "come down to the size of a football, and probably operate 10
times faster.''

Statements like that overwhelm me. I struggle to grasp their
meaning. Several years ago, when supercomputers were arriving on the
scene, I read that one of them could do 4 million "long''
multiplications in a second. In an attempt to give some relevance to
that statement, I decided to figure out how long it would take me to do
that many multiplications, using the old-fashioned paper-and-pencil
method I learned in school. I'm sure, for the computer, a "long''
multiplication was more than multiplying a five-digit number by a
four-digit number, but that was "long'' by my standards. I timed myself
and found I could do such a multiplication, with reasonable care, in
about 75 seconds. At that rate, it would take me 300 million seconds to
do 4 million muliplications. That's 83,333 hours. If I limited myself
to an eight-hour workday, five days a week, 50 weeks a year, it would
take me about 42 years. That's a lifetime's work, and a supercomputer
does it in one second! Now I'm told, the next generation of computers
will do 10 lifetimes' work in a second.

So what's my reaction? It ranges from amazement and celebration to
peevishness. The advances in computational tools I have used in my
lifetime astound me--from the paper and pencil of school arithmetic,
the slide rule of high-school physics, and the hand-cranked calculating
machine of college statistics, to those marvelous tools, the
solar-powered calculator and personal computer--and there's more to
come. In my wildest schoolboy imagination, I would never have dreamed
of possessing such powerful and tractable computing devices. But I
do--and when I think of all the time and effort they save me and the
possibilities they afford me, I revel in their use.

So what peeves me? Schoolchildren aren't allowed to join in the
celebration. Despite all these magnificent advancements that have
brought untold computational power to one's fingertips, school is "more
like it's been than it's ever been before.'' As in the days of my
youth, and my grandparents' youth, schoolchildren are drilled for hours
to perform paper-and-pencil computations in machine-like fashion. There
are no breakthroughs here. There are no superconductors that have
managed to overcome the resistance that will allow any electricity to
permeate the mathematics classroom.

Instead, there are staunch defenders of the status quo. Those that
insist on turning students into paper-and-pencil computing machines.
For what purpose? It's ridiculous to expect there is a future for a
paper-and-pencil machine that takes a lifetime to do what another
machine can do in a moment. It's no longer a matter of economics when a
hand-held calculator costs little more than a good mechanical pencil.
And solar power has stilled those ominous voices that warned us about
batteries going dead. So why insist on centering a mathematics
curriculum on teaching students how to use outmoded computational
tools?

The ostensible reason proffered by those resisting change is that
students must learn the "basic mathematical skills.'' And, in order to
do that, the use of electronic tools must be curtailed or even banned.
As the author of one current textbook series explained, in a national
advertisement, "... calculators should not be permitted in elementary
schools, for this is the time and the place for learning fundamental
concepts and mastering paper-and-pencil skills. If students are
permitted to use calculators too early, many of them will block and
refuse to do the drudgery necessary to perfect the necessary
paper-and-pencil skills.''

Such cries for skill drudgery come close to skullduggery. In the
first place, paper-and-pencil skills are not "necessary.'' Secondly,
anything that can remove drudgery from school mathematics programs
ought to be extolled instead of condemned.

Designating a mathematical skill as "necessary'' implies that it is
needed to function mathematically. That is not the case with
paper-and-pencil arithmetical skills; one can function mathematically
quite well without them. You may object, pointing out that one can't
get through school mathematics without them. That is very likely true,
but that doesn't mean they are basic mathematical skills. That only
means they are school-survival skills.

To determine whether or not a mathematical skill is necessary, one
ought to examine its essentialness in the nonschool parts of the world.
Over the half-century I have been doing mathematics--as a schoolboy, as
a college and graduate student, in any number of odd jobs that paid my
way though school, as an industrial mathematician, as a university
teacher and researcher, in everyday life, and just for fun--there is
nothing I have done, apart from schoolwork, that today requires the use
of paper-and-pencil arithmetical procedures. Calculators provide an
economical and efficient way of doing computations I can't do in my
head. And knowledge of these paper-and-pencil procedures does not
provide me with mathematical insight of any significance.

There are times when I find these paper-and-pencil computational
skills useful--although I can't remember the last time I used them for
long division other than at school. Also, there are those who prefer
these methods of computation. But even if these procedures are
occasionally useful, or preferred by some, it does not follow that they
are necessary skills. In my mathematical life, I can get along without
them. Most adults do.

This doesn't only apply to the paper-and-pencil procedures of
elementary-school arithmetic. It also applies to the paper-and-pencil
procedures of high-school algebra, college calculus, and all other math
courses. These days, any step-by-step procedure involving the
manipulation of mathematical symbols, according to a fixed set of
rules, can be done by a calculator or computer. Some procedures are
simple enough that they are best done mentally or by hand, but any that
require more than a modicum of time and energy to do manually are most
economically done by machine. And, outside school, they are.

Thus, school math programs that center on the mastery of mechanical
paper-and-pencil procedures are not necessary skills. They are vestiges
of another age, when human beings, in conjunction with paper and
pencil, were the computing machines of the day. To gear a math program
to producing such machines does indeed reduce students to drudges.

I suspect the resistance to calculators in classrooms is not a
tenacity for teaching basic skills, but rather an anxiety about what to
do if existing programs are abandoned. I suspect many educators share
the feelings of that 5th-grade teacher whose immediate response to the
suggestion that he allow calculators to be freely used in his classroom
was, "But that would destroy my whole program!'' It would. However,
once one sees the truth of that statement, lets the initial shock wear
off, and asks what ought to happen next, one can envision a mathematics
program that recognizes current technology, is economically feasible,
and provides pertinent mathematics for purposeful students, without
drudgery.

Such a program does not require that classrooms be equipped with the
latest in electronic computing devices. Rather, it requires that the
existence of these devices be recognized, and time and energy not be
wasted teaching students pencil-and-paper procedures that, except in
school, are done electronically. For most school computational
purposes, inexpensive calculators will do. And, since calculators are
easy to use, math programs needs not devote much attention to
computational skills.

Thus, computation plays a minor role in a pertinent math program.
Such a program will emphasize meaning rather than symbolic
manipulation. It will educe the mathematical creativity innate in every
student; it will develop mathematical insight and intuition; it will
stress cooperative problem-solving; and it will allow students to
compute by whatever means they can--mentally, counting on their
fingers, with an abacus, using pencil and paper, or punching the keys
of a calculator. As students grow in mathematical maturity, they will
find the computational methods that work best for them.

I recognize that there is a vast difference between listing
characteristics of a math program that is appropriate for the
electronic age and implementing such a program in the schools. Doing
the latter is as exciting and challenging as searching for
superconductors.

For the past several years, I have been involved with a coalition of
school and college math educators who are working to instill the above
characteristics into portions of the school mathematics program. The
vehicle we have chosen is visual thinking--the use of sensory
perception, models, sketches, and imagery to provide insight into
mathematical concepts and bring meaning to mathematical symbolism. It's
gratifying to watch general-mathematics students--who have become
accustomed to perfunctory paper-and-pencil drill carried out with
little meaning, mediocre success, and no interest--make contact with
their mathematical instincts and come alive mathematically. It's
rewarding to see math-anxious elementary teachers overcome their doubts
about ever understanding mathematics, or tackling an open-ended
mathematical question that they cannot solve mechanically. And it's
encouraging to know they no longer will pass on an apprehensive and
distorted view of mathematics to their students.

There are other people, scattered throughout the country, engaged in
similar activities. These are the people who celebrate calculators and
computers for the computational power they bring to all students. They
are in contact with their own mathematical spirit, ignite the
mathematical spark in others, and know the essentials needed to nourish
it. These are the people who, despite my peevishness, give me hope. It
is their energy that can overcome the resistance of those who impede
mathematical power with the drudgery of mechanical paper-and-pencil
drill. It is their energy, conducted into classrooms, that can
electrify the mathematical potential inherent in every student.

Eugene A. Maier is a professor of mathematical sciences at Portland
State University, and president of the Math Learning Center, Salem,
Ore.

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